A few Thing

First of all - why no native Zimbra Support - why this must have Sync?
only to not enable OSS full support? Sorry this is poor and doesnt really make sense.

A full native App would be way better - then you can even cancle the Excahnge Support - active sync has to many issues.

Besides that now lets begin at Start.
I know you might wanna build a ALl In One App - as nice at it is on Desktop as useless it is on Mobile many Times.

SImply because the way things are used in the real World are so different.
On Mobile many people used it while walk or in the Subway - mostly with one hand - it has to be quick and easy - important things have to be the fokus

switching to the kalendar over the folders or the small dropdown left upper corner - man this isnt good.

i would suggest a way different way -
first all app inside - calendar - mail - tasks.... should have the ability to start alone and very very quick with their own icons
to have all integrated - in addition a little zimbra dashboard
where you got them all in one - i think it was roadsync who had the nicest dashboard
8 or 10 icons i think in that board and mainsettings

also the applications itself are real lacking of options

i would suggest to look a references:
task - tasksync for excahnge
mail - k9mail or moxier - only downside at k9 is that folder thing (btw you made the folders very nice thats a plus)

calendar - finally - take a look at "acalendar" its also free and the best on the market atm - very fast very native useability

i think most important is useability - getting things done thing .. quick - dont let us look 20 sec to find the option we need
but also let us configure which options/things we need

btw would be nice to have a cos for those options later

if you want i can make you a complete lost of options im missing most

ah yea last but not least contacts.
ahm im right you sync only zimbra contacts to the application - no phone contacts? or im wrong?
if so sync the phone and use the phone for lookups would be better

ah yea and find a way to use qr9 codes where it make sense
this is also usefull for zimbra itself - they are now everywhere

Soap support

Syncing mail, tasks, and events is fairly ok via ActiveSync. However, syncing contacts is fairly limited via ActiveSync. Creating a new SyncAdapter utilising soap for Contact syncing directly to the Android contact store would be a far better way forward, and ensure that custom fields, and multiple fields would be synced, and no data lost.